Sunday Afternoon Maths VII Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Ninety Nine

Every item bought will cause the pence in the total cost to fall by 1. So to spend £65.76, Susanna must have bought 24 items.

Extension

What is the smallest amount Susanna could spend for which we could not tell how many items she bought?

Reverse Bases

If ab in base 10 is equal to ba in base 4, then 10a+b=4b+a.

So, 9a=3b.

a and b must both be less than 4, as they are digits used in base 4, so a=1 and b=3.

So 13 in base 10 is equal to 31 in base 4.

By the same method, we find that:
23 in base 10 is equal to 32 in base 7
46 in base 10 is equal to 64 in base 7.

12 in base 9 is equal to 21 in base 5.
24 in base 9 is equal to 42 in base 5.

Extension

For which pairs of bases A and B can you find two digits g and h such that gh in base A is equal to hg in base B?